IE582 - Fall2021

HOMEWORK 1

TASK 1

D=(1:15)

set.seed(582)
dim1=array(runif(1000*D[1],-1,1), dim=c(1000, D[1]))
dim2=array(runif(1000*D[2],-1,1), dim=c(1000,D[2]))
dim3=array(runif(1000*D[3],-1,1),dim=c(1000,D[3]))
dim4=array(runif(1000*D[4],-1,1), dim=c(1000,D[4]))
dim5=array(runif(1000*D[5],-1,1), dim=c(1000,D[5]))
dim6=array(runif(1000*D[6],-1,1), dim=c(1000,D[6]))
dim7=array(runif(1000*D[7],-1,1), dim=c(1000,D[7]))
dim8=array(runif(1000*D[8],-1,1), dim=c(1000,D[8]))
dim9=array(runif(1000*D[9],-1,1), dim=c(1000,D[9]))
dim10=array(runif(1000*D[10],-1,1), dim=c(1000,D[10]))
dim11=array(runif(1000*D[11],-1,1), dim=c(1000,D[11]))
dim12=array(runif(1000*D[12],-1,1), dim=c(1000,D[12]))
dim13=array(runif(1000*D[13],-1,1), dim=c(1000,D[13]))
dim14=array(runif(1000*D[14],-1,1), dim=c(1000,D[14]))
dim15=array(runif(1000*D[15],-1,1), dim=c(1000,D[15]))

A)

distance1=array()
for (i in 1:nrow(dim1)){
    sum=0
    for(j in 1:ncol(dim1)){
    sum = sum + (dim1[i,j]**2)
    }
    distance1[i] = sqrt(sum)
}

distance2=array()
for (i in 1:nrow(dim2)){
    sum=0
    for(j in 1:ncol(dim2)){
    sum = sum + (dim2[i,j]**2)
    }
    distance2[i] = sqrt(sum)
}

distance3=array()
for (i in 1:nrow(dim3)){
    sum=0
    for(j in 1:ncol(dim3)){
    sum = sum + (dim3[i,j]**2)
    }
    distance3[i] = sqrt(sum)
}

distance4=array()
for (i in 1:nrow(dim4)){
    sum=0
    for(j in 1:ncol(dim4)){
    sum = sum + (dim4[i,j]**2)
    }
    distance4[i] = sqrt(sum)
}

distance5=array()
for (i in 1:nrow(dim5)){
    sum=0
    for(j in 1:ncol(dim5)){
    sum = sum + (dim5[i,j]**2)
    }
    distance5[i] = sqrt(sum)
}

distance6=array()
for (i in 1:nrow(dim6)){
    sum=0
    for(j in 1:ncol(dim6)){
    sum = sum + (dim6[i,j]**2)
    }
    distance6[i] = sqrt(sum)
}

distance7=array()
for (i in 1:nrow(dim7)){
    sum=0
    for(j in 1:ncol(dim7)){
    sum = sum + (dim7[i,j]**2)
    }
    distance7[i] = sqrt(sum)
}

distance8=array()
for (i in 1:nrow(dim8)){
    sum=0
    for(j in 1:ncol(dim8)){
    sum = sum + (dim8[i,j]**2)
    }
    distance8[i] = sqrt(sum)
}

distance9=array()
for (i in 1:nrow(dim9)){
    sum=0
    for(j in 1:ncol(dim9)){
    sum = sum + (dim9[i,j]**2)
    }
    distance9[i] = sqrt(sum)
}

distance10=array()
for (i in 1:nrow(dim10)){
    sum=0
    for(j in 1:ncol(dim10)){
    sum = sum + (dim10[i,j]**2)
    }
    distance10[i] = sqrt(sum)
}

distance11=array()
for (i in 1:nrow(dim11)){
    sum=0
    for(j in 1:ncol(dim11)){
    sum = sum + (dim11[i,j]**2)
    }
    distance11[i] = sqrt(sum)
}

distance12=array()
for (i in 1:nrow(dim12)){
    sum=0
    for(j in 1:ncol(dim12)){
    sum = sum + (dim12[i,j]**2)
    }
    distance12[i] = sqrt(sum)
}

distance13=array()
for (i in 1:nrow(dim13)){
    sum=0
    for(j in 1:ncol(dim13)){
    sum = sum + (dim13[i,j]**2)
    }
    distance13[i] = sqrt(sum)
}

distance14=array()
for (i in 1:nrow(dim14)){
    sum=0
    for(j in 1:ncol(dim14)){
    sum = sum + (dim14[i,j]**2)
    }
    distance14[i] = sqrt(sum)
}

distance15=array()
for (i in 1:nrow(dim15)){
    sum=0
    for(j in 1:ncol(dim15)){
    sum = sum + (dim15[i,j]**2)
    }
    distance15[i] = sqrt(sum)
}

distances = matrix(c(distance1,distance2,distance3,
                 distance4,distance5,distance6,distance7,
                 distance8,distance9,distance10,distance11,
                 distance12,distance13,distance14,distance15), nrow=1000,ncol=15 )
percentages=array()

for(i in 1:ncol(distances)){
    percentages[i]= sum(distances[,i]<1)/1000
}

Fractions of points within distance 1

percentages

1. 1
2. 0.799
3. 0.524
4. 0.319
5. 0.145
6. 0.073
7. 0.029
8. 0.018
9. 0.006
10. 0.002
11. 0
12. 0
13. 0
14. 0
15. 0

plot(y=percentages, x=c(1:15), main="Fraction of Points Within Distance 1",xlab = "Dimension",
     ylab="Fraction")

B)

For D=2 The fraction approximates the ratio of the volume of the unit circle to the square inscribing it.

The area of the circle is calculated as:

(Π*r^2)

The area of the square inscribing the circle with r=1 is 4. So;

0.799 = (Π*r^2) / 4 where r=1

Π is approximately calculated as:

pi_approx_dim2 = 0.799*4
pi_approx_dim2

3.196

For D=3 The fraction approximates the ratio of the volume of the unit sphere to the cube inscribing it.

The volume of the circle is calculated as:

(4/3)Πr^3

The volume of the cube inscribing the sphere with r=1 is 8. So;

0.524 = (4/3)Πr^3 / 8 where r=1

Π is approximately calculated as:

pi_approx_dim3 = 0.524*6
pi_approx_dim3

3.144

C)

To observe the approximation of Π I will perform the calculations for larger sample sizes for 2 dimensional plane and 3 dimensional space.

For D=2

set.seed(58)
dim2_5000=array(runif(5000*D[2],-1,1), dim=c(5000,D[2]))
dim2_10000=array(runif(10000*D[2],-1,1), dim=c(10000,D[2]))
dim2_25000=array(runif(25000*D[2],-1,1), dim=c(25000,D[2]))
dim2_100000=array(runif(100000*D[2],-1,1), dim=c(100000,D[2]))
dim2_50000=array(runif(50000*D[2],-1,1), dim=c(50000,D[2]))

distance2_5000=array()
for (i in 1:nrow(dim2_5000)){
  sum=0
  for(j in 1:ncol(dim2_5000)){
    sum = sum + (dim2_5000[i,j]**2)
  }
  distance2_5000[i] = sqrt(sum)
}

percentages_2d=array()
percentages_2d[1] = percentages[2]
percentages_2d[2]= sum(distance2_5000<1)/5000


distance2_10000=array()
for (i in 1:nrow(dim2_10000)){
  sum=0
  for(j in 1:ncol(dim2_10000)){
    sum = sum + (dim2_10000[i,j]**2)
  }
  distance2_10000[i] = sqrt(sum)
}

percentages_2d[3]= sum(distance2_10000<1)/10000



distance2_25000=array()
for (i in 1:nrow(dim2_25000)){
  sum=0
  for(j in 1:ncol(dim2_25000)){
    sum = sum + (dim2_25000[i,j]**2)
  }
  distance2_25000[i] = sqrt(sum)
}

percentages_2d[4]= sum(distance2_25000<1)/25000



distance2_50000=array()
for (i in 1:nrow(dim2_50000)){
  sum=0
  for(j in 1:ncol(dim2_50000)){
    sum = sum + (dim2_50000[i,j]**2)
  }
  distance2_50000[i] = sqrt(sum)
}
percentages_2d[5]= sum(distance2_50000<1)/50000



distance2_100000=array()
for (i in 1:nrow(dim2_100000)){
  sum=0
  for(j in 1:ncol(dim2_100000)){
    sum = sum + (dim2_100000[i,j]**2)
  }
  distance2_100000[i] = sqrt(sum)
}

percentages_2d[6]= sum(distance2_100000<1)/100000

pi_approx_2d = percentages_2d*4

plot(y=pi_approx_2d, x=c(1,5,10,25,50,100),main="Approximation of pi for D=2",xlab = "Sample Size(*1000)",
     ylab="Fraction")

For D=3

set.seed(58)
dim3_5000=array(runif(5000*D[3],-1,1), dim=c(5000,D[3]))
dim3_10000=array(runif(10000*D[3],-1,1), dim=c(10000,D[3]))
dim3_25000=array(runif(25000*D[3],-1,1), dim=c(25000,D[3]))
dim3_100000=array(runif(100000*D[3],-1,1), dim=c(100000,D[3]))
dim3_50000=array(runif(50000*D[3],-1,1), dim=c(50000,D[3]))

distance3_5000=array()
for (i in 1:nrow(dim3_5000)){
  sum=0
  for(j in 1:ncol(dim3_5000)){
    sum = sum + (dim3_5000[i,j]**2)
  }
  distance3_5000[i] = sqrt(sum)
}

percentages_3d=array()
percentages_3d[1] = percentages[3]
percentages_3d[2]= sum(distance3_5000<1)/5000


distance3_10000=array()
for (i in 1:nrow(dim3_10000)){
  sum=0
  for(j in 1:ncol(dim3_10000)){
    sum = sum + (dim3_10000[i,j]**2)
  }
  distance3_10000[i] = sqrt(sum)
}

percentages_3d[3]= sum(distance3_10000<1)/10000



distance3_25000=array()
for (i in 1:nrow(dim3_25000)){
  sum=0
  for(j in 1:ncol(dim3_25000)){
    sum = sum + (dim3_25000[i,j]**2)
  }
  distance3_25000[i] = sqrt(sum)
}

percentages_3d[4]= sum(distance3_25000<1)/25000



distance3_50000=array()
for (i in 1:nrow(dim3_50000)){
  sum=0
  for(j in 1:ncol(dim3_50000)){
    sum = sum + (dim3_50000[i,j]**2)
  }
  distance3_50000[i] = sqrt(sum)
}
percentages_3d[5]= sum(distance3_50000<1)/50000



distance3_100000=array()
for (i in 1:nrow(dim3_100000)){
  sum=0
  for(j in 1:ncol(dim3_100000)){
    sum = sum + (dim3_100000[i,j]**2)
  }
  distance3_100000[i] = sqrt(sum)
}

percentages_3d[6]= sum(distance3_100000<1)/100000

pi_approx_3d = percentages_3d*6

plot(y=pi_approx_3d, x=c(1,5,10,25,50,100),main="Approximation of pi for D=3",xlab = "Sample Size(*1000)",
     ylab="Fraction")

For both D=2 and D=3 as the sample size gets larger approximation of pi gets closer to the actual value. Fluctutations of the approximated values around the real value is expected as a result of randomness.

For D=3 the approximation is very close to the real value when the sample size is 1000.This can also be considered as a result of randomness. As the sample size gets larger approximation shows the expected behavior.

D) NEAREST NEIGHBORS

set.seed(52)
dim1_test=array(runif(100*D[1],-1,1), dim=c(100, D[1]))
dim2_test=array(runif(100*D[2],-1,1), dim=c(100,D[2]))
dim3_test=array(runif(100*D[3],-1,1),dim=c(100,D[3]))
dim4_test=array(runif(100*D[4],-1,1), dim=c(100,D[4]))
dim5_test=array(runif(100*D[5],-1,1), dim=c(100,D[5]))
dim6_test=array(runif(100*D[6],-1,1), dim=c(100,D[6]))
dim7_test=array(runif(100*D[7],-1,1), dim=c(100,D[7]))
dim8_test=array(runif(100*D[8],-1,1), dim=c(100,D[8]))
dim9_test=array(runif(100*D[9],-1,1), dim=c(100,D[9]))
dim10_test=array(runif(100*D[10],-1,1), dim=c(100,D[10]))
dim11_test=array(runif(100*D[11],-1,1), dim=c(100,D[11]))
dim12_test=array(runif(100*D[12],-1,1), dim=c(100,D[12]))
dim13_test=array(runif(100*D[13],-1,1), dim=c(100,D[13]))
dim14_test=array(runif(100*D[14],-1,1), dim=c(100,D[14]))
dim15_test=array(runif(100*D[15],-1,1), dim=c(100,D[15]))

neighbors1=matrix(,100,1000)

for(k in 1:nrow(dim1_test)){
    for (i in 1:nrow(dim1)){
      sum=0
      for(j in 1:ncol(dim1)){
          sum = sum +  ((dim1_test[k,j]-dim1[i,j])**2)
      }
      neighbors1[k,i] = sqrt(sum)
    }
}

nearest_neighbors1 = array()
for(i in 1:nrow(neighbors1)){
    nearest_neighbors1[i]=min(neighbors1[i,])
}

neighbors2=matrix(,100,1000)

for(k in 1:nrow(dim2_test)){
    for (i in 1:nrow(dim2)){
      sum=0
      for(j in 1:ncol(dim2)){
          sum = sum +  ((dim2_test[k,j]-dim2[i,j])**2)
      }
      neighbors2[k,i] = sqrt(sum)
    }
}

nearest_neighbors2 = array()
for(i in 1:nrow(neighbors2)){
    nearest_neighbors2[i]=min(neighbors2[i,])
}

neighbors3=matrix(,100,1000)

for(k in 1:nrow(dim3_test)){
    for (i in 1:nrow(dim3)){
      sum=0
      for(j in 1:ncol(dim3)){
          sum = sum +  ((dim3_test[k,j]-dim3[i,j])**2)
      }
      neighbors3[k,i] = sqrt(sum)
    }
}

nearest_neighbors3 = array()
for(i in 1:nrow(neighbors3)){
    nearest_neighbors3[i]=min(neighbors3[i,])
}

neighbors4=matrix(,100,1000)

for(k in 1:nrow(dim4_test)){
    for (i in 1:nrow(dim4)){
      sum=0
      for(j in 1:ncol(dim4)){
          sum = sum +  ((dim4_test[k,j]-dim4[i,j])**2)
      }
      neighbors4[k,i] = sqrt(sum)
    }
}

nearest_neighbors4 = array()
for(i in 1:nrow(neighbors4)){
    nearest_neighbors4[i]=min(neighbors4[i,])
}

neighbors5=matrix(,100,1000)

for(k in 1:nrow(dim5_test)){
    for (i in 1:nrow(dim5)){
      sum=0
      for(j in 1:ncol(dim5)){
          sum = sum +  ((dim5_test[k,j]-dim5[i,j])**2)
      }
      neighbors5[k,i] = sqrt(sum)
    }
}

nearest_neighbors5 = array()
for(i in 1:nrow(neighbors5)){
    nearest_neighbors5[i]=min(neighbors5[i,])
}

neighbors6=matrix(,100,1000)

for(k in 1:nrow(dim6_test)){
    for (i in 1:nrow(dim6)){
      sum=0
      for(j in 1:ncol(dim6)){
          sum = sum +  ((dim6_test[k,j]-dim6[i,j])**2)
      }
      neighbors6[k,i] = sqrt(sum)
    }
}

nearest_neighbors6 = array()
for(i in 1:nrow(neighbors6)){
    nearest_neighbors6[i]=min(neighbors6[i,])
}

neighbors7=matrix(,100,1000)

for(k in 1:nrow(dim7_test)){
    for (i in 1:nrow(dim7)){
      sum=0
      for(j in 1:ncol(dim7)){
          sum = sum +  ((dim7_test[k,j]-dim7[i,j])**2)
      }
      neighbors7[k,i] = sqrt(sum)
    }
}

nearest_neighbors7 = array()
for(i in 1:nrow(neighbors7)){
    nearest_neighbors7[i]=min(neighbors7[i,])
}

neighbors8=matrix(,100,1000)

for(k in 1:nrow(dim8_test)){
    for (i in 1:nrow(dim8)){
      sum=0
      for(j in 1:ncol(dim8)){
          sum = sum +  ((dim8_test[k,j]-dim8[i,j])**2)
      }
      neighbors8[k,i] = sqrt(sum)
    }
}

nearest_neighbors8 = array()
for(i in 1:nrow(neighbors8)){
    nearest_neighbors8[i]=min(neighbors8[i,])
}

neighbors9=matrix(,100,1000)

for(k in 1:nrow(dim9_test)){
    for (i in 1:nrow(dim9)){
      sum=0
      for(j in 1:ncol(dim9)){
          sum = sum +  ((dim9_test[k,j]-dim9[i,j])**2)
      }
      neighbors9[k,i] = sqrt(sum)
    }
}

nearest_neighbors9 = array()
for(i in 1:nrow(neighbors9)){
    nearest_neighbors9[i]=min(neighbors9[i,])
}

neighbors10=matrix(,100,1000)

for(k in 1:nrow(dim10_test)){
    for (i in 1:nrow(dim10)){
      sum=0
      for(j in 1:ncol(dim10)){
          sum = sum +  ((dim10_test[k,j]-dim10[i,j])**2)
      }
      neighbors10[k,i] = sqrt(sum)
    }
}

nearest_neighbors10 = array()
for(i in 1:nrow(neighbors10)){
    nearest_neighbors10[i]=min(neighbors10[i,])
}

neighbors11=matrix(,100,1000)

for(k in 1:nrow(dim11_test)){
    for (i in 1:nrow(dim11)){
      sum=0
      for(j in 1:ncol(dim11)){
          sum = sum +  ((dim11_test[k,j]-dim11[i,j])**2)
      }
      neighbors11[k,i] = sqrt(sum)
    }
}

nearest_neighbors11 = array()
for(i in 1:nrow(neighbors11)){
    nearest_neighbors11[i]=min(neighbors11[i,])
}

neighbors12=matrix(,100,1000)

for(k in 1:nrow(dim12_test)){
    for (i in 1:nrow(dim12)){
      sum=0
      for(j in 1:ncol(dim12)){
          sum = sum +  ((dim12_test[k,j]-dim12[i,j])**2)
      }
      neighbors12[k,i] = sqrt(sum)
    }
}

nearest_neighbors12= array()
for(i in 1:nrow(neighbors12)){
    nearest_neighbors12[i]=min(neighbors12[i,])
}

neighbors13=matrix(,100,1000)

for(k in 1:nrow(dim13_test)){
    for (i in 1:nrow(dim13)){
      sum=0
      for(j in 1:ncol(dim13)){
          sum = sum +  ((dim13_test[k,j]-dim13[i,j])**2)
      }
      neighbors13[k,i] = sqrt(sum)
    }
}

nearest_neighbors13 = array()
for(i in 1:nrow(neighbors13)){
    nearest_neighbors13[i]=min(neighbors13[i,])
}

neighbors14=matrix(,100,1000)

for(k in 1:nrow(dim14_test)){
    for (i in 1:nrow(dim14)){
      sum=0
      for(j in 1:ncol(dim14)){
          sum = sum +  ((dim14_test[k,j]-dim14[i,j])**2)
      }
      neighbors14[k,i] = sqrt(sum)
    }
}

nearest_neighbors14 = array()
for(i in 1:nrow(neighbors14)){
    nearest_neighbors14[i]=min(neighbors14[i,])
}

neighbors15=matrix(,100,1000)

for(k in 1:nrow(dim15_test)){
    for (i in 1:nrow(dim15)){
      sum=0
      for(j in 1:ncol(dim15)){
          sum = sum +  ((dim15_test[k,j]-dim15[i,j])**2)
      }
      neighbors15[k,i] = sqrt(sum)
    }
}

nearest_neighbors15 = array()
for(i in 1:nrow(neighbors15)){
    nearest_neighbors15[i]=min(neighbors15[i,])
}

avg_distances = c(mean(nearest_neighbors1),mean(nearest_neighbors2),mean(nearest_neighbors3),
mean(nearest_neighbors4),mean(nearest_neighbors5),mean(nearest_neighbors6),mean(nearest_neighbors7),
mean(nearest_neighbors8),mean(nearest_neighbors9),mean(nearest_neighbors10),mean(nearest_neighbors11),
mean(nearest_neighbors12),mean(nearest_neighbors13),mean(nearest_neighbors14),mean(nearest_neighbors15))

plot(y=avg_distances, x=c(1:15),main="Average Nearest Neighbor Distances",xlab = "Dimension",
     ylab="Distance")

TASK 2

require(jpeg)
setwd("C:/Users/Pınar YILDIRIM/Desktop")

Loading required package: jpeg

my_image=readJPEG("homework1.jpg")

The image is read with the readJPEG function and stored as my_image. As stated below the object is numeric array, which consists of "doubles", with dimensions 512x512x3. The results are consistent with the information given in the homework description. There are 3 512x512 matrices to represent the color intensity of each pixel for 3 colors RED, GREEN and BLUE.

is.numeric(my_image)
is.character(my_image)
typeof(my_image)
class(my_image)
dim(my_image)
str(my_image)

TRUE

FALSE

'double'

'array'

1. 512
2. 512
3. 3

 num [1:512, 1:512, 1:3] 0.796 0.792 0.796 0.804 0.808 ...

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((my_image), 100, 400, 150, 450)

When matrices representing each channel is displayed the result is a black and white image that only demonstrates the instensity of the colors.

plot(c(100, 400), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((my_image[,,1]), 100, 400, 200, 450)
rasterImage((my_image[,,2]), 200, 400, 300, 450)
rasterImage((my_image[,,3]), 300, 400, 400, 450)

To observe colors I created a matrix of zeros with size 512x512. Then I created 3 arrays with sizes 512x512x3. I assigned each color red, blue and green to the corresponding space of one matrix and filled remaining spaces with the zero matrices.

zeros=array(rep(0,(512*512)),dim=c(512,512))

red=array(c(my_image[,,1],zeros,zeros),dim=c(512,512,3))
green=array(c(zeros,my_image[,,2],zeros),dim=c(512,512,3))
blue=array(c(zeros,zeros,my_image[,,3]),dim=c(512,512,3))

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((red), 100, 400, 150, 450)
plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((green), 100, 400, 150, 450)
plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((blue), 100, 400, 150, 450)

When I put the related elements of matrices red, blue and green together and discard the zeros I get the original image.

comp=array(c(red[,,1],green[,,2],blue[,,3]),dim=c(512,512,3))

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((comp), 100, 400, 150, 450)

For each channel column means are calculated and plotted as below.

r = my_image[,,1]
g = my_image[,,2]
b = my_image[,,3]

r_colmeans =array()

for (i in 1:512){
    r_colmeans[i] = mean(r[,i])
}

g_colmeans =array()

for (i in 1:512){
    g_colmeans[i] = mean(g[,i])
}


b_colmeans =array()

for (i in 1:512){
    b_colmeans[i] = mean(b[,i])
}

require(ggplot2)

(ggplot() + geom_line(aes(x=c(1:512), y=r_colmeans, color="red"))
  +geom_line(aes(x=c(1:512), y=g_colmeans, color="green"))
  +geom_line(aes(x=c(1:512), y=b_colmeans, color="blue"))
 + labs(title = "Column Means", x="Column", y="Mean")
)

Loading required package: ggplot2
Registered S3 methods overwritten by 'ggplot2':
  method         from 
  [.quosures     rlang
  c.quosures     rlang
  print.quosures rlang

I split the image into two halves and subtracted the right half from the left. The two halves and new image is shown below negative pixel values are set to zero:

r_left = r[,1:256]
r_right = r[,512:257]
r_new=r_left-r_right

for (i in 1:length(r_new)){
    if(r_new[i]<0){
        r_new[i] = 0
    }
}


g_left = g[,1:256]
g_right = g[,512:257]
g_new=g_left-g_right

for (i in 1:length(g_new)){
    if(g_new[i]<0){
        g_new[i] = 0
    }
}


b_left = b[,1:256]
b_right = b[,512:257]
b_new=b_left-b_right

for (i in 1:length(b_new)){
    if(b_new[i]<0){
        b_new[i] = 0
    }
}

left =array(c(r_left,g_left,b_left),dim=c(512,256,3)) 
right =array(c(r_right,g_right,b_right),dim=c(512,256,3)) 
comp_new=array(c(r_new,g_new,b_new),dim=c(512,256,3))

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((left), 100, 400, 150, 450)

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((right), 100, 400, 150, 450)

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((comp_new), 100, 400, 150, 450)





plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((b_left), 100, 400, 150, 450)

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((b_right), 100, 400, 150, 450)

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((b_new), 100, 400, 150, 450)

Red channel of new image is displayed below:

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((r_left), 100, 400, 150, 450)

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((r_right), 100, 400, 150, 450)

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((r_new), 100, 400, 150, 450)

zeros_new=array(rep(0,(512*256)),dim=c(512,256))

red_new=array(c(r_new,zeros_new,zeros_new),dim=c(512,256,3))

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((red_new), 100, 400, 150, 450)

Green channel of new image is displayed below:

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((g_left), 100, 400, 150, 450)

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((g_right), 100, 400, 150, 450)

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((g_new), 100, 400, 150, 450)

green_new=array(c(zeros_new,g_new,zeros_new),dim=c(512,256,3))

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((green_new), 100, 400, 150, 450)

Blue channel of new image is displayed below:

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((b_left), 100, 400, 150, 450)

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((b_right), 100, 400, 150, 450)

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((b_new), 100, 400, 150, 450)

blue_new=array(c(zeros_new,zeros_new,b_new),dim=c(512,256,3))

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((blue_new), 100, 400, 150, 450)

To create a noisy image noise is added to the each pixel of the original image. Noise values are generated from uniform distribution with minimum value of 0 and a maximum value of “0.1 * maximum pixel value observed” to each pixel value for each channel of original image.

Maximum pixel value observed is 1 and there are 47 pixels with value 1.

max(my_image)
sum(my_image==1)

1

47

my_image_noisy= array(,dim=c(512,512,3))

set.seed(23)
for(i in 1:3){
    for(j in 1:512){
        for(k in 1:512){
            my_image_noisy[k,j,i] = my_image[k,j,i]+runif(1,0,0.1)
        }
    }
}

for(i in 1:3){
    for(j in 1:512){
        for(k in 1:512){
            if(my_image_noisy[k,j,i] > 1){  
            my_image_noisy[k,j,i] = 1
            }
        }
    }
}

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((my_image_noisy), 100, 400, 150, 450)

plot(c(100, 400), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((my_image_noisy[,,1]), 100, 400, 200, 450)
rasterImage((my_image_noisy[,,2]), 200, 400, 300, 450)
rasterImage((my_image_noisy[,,3]), 300, 400, 400, 450)

zeros=array(rep(0,(512*512)),dim=c(512,512))

red_noisy=array(c(my_image_noisy[,,1],zeros,zeros),dim=c(512,512,3))
green_noisy=array(c(zeros,my_image_noisy[,,2],zeros),dim=c(512,512,3))
blue_noisy=array(c(zeros,zeros,my_image_noisy[,,3]),dim=c(512,512,3))

plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((red_noisy), 100, 400, 150, 450)
plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((green_noisy), 100, 400, 150, 450)
plot(c(100, 150), c(400, 450), type = "n", xlab = "", ylab = "")
rasterImage((blue_noisy), 100, 400, 150, 450)